You are a casino owner, gaming manager or surveillance manager. You have a feeling that something is going down at the roulette tables, but you can’t put your finger on it. Looking for a method to calculate the probabilities?

Excelpunks puts the statistics into this equation with Roulette Analysis!

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•Round-By-Round Analysis and Entire Game Analysis:

Occurrence of Specific numbers

Win/Loss analysis on all wager types

Win/Loss analysis on Specific Numbers

Win/Loss analysis of the game as a whole

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Let robust statistical principles do the math for you. Think that your player is winning more than he should? Let Roulette Analysis prove it mathematically!

The central limit theorem proposes that up to 99.9% of all occurrences happen between -3 to -1 and 1 to 3 σs from the average or mean. σ extends into the positive (meaning 1 to 3 σ) and negative (meaning -1 to -3 σ).

In order to derive any sort of boundary in a casino game, one has to calculate the following:

Mean Or Average Probability Of An Event Occurring (this is to determine the average expected result)

Standard Deviation Of The Event Occurring (this is to determine σ)

Probability Of Events Occurring

The probability of events refers to the mathematical probability of an event. This can be calculated by determining the possible outcomes of what you are trying to measure and then dividing that by the total possible outcomes for an event.

A simple example would involve a single deck of cards. The probability of drawing an Ace of Spades is 1/52, since there is only one Ace of Spades in the deck. The probability of drawing an Ace of any suit is 4/52, since there are four Aces in the deck.

Mean Or Average Probability Of An Event Occurring

For this, we turn to binomial distribution. Binomial distribution calculates the means and standard deviations of occurrences that have one of two outcomes.

Example: Baccarat

Baccarat is a good example, where the outcome is either a Player or Banker. For simplicity, let’s disregard coups ending in a tie for now.

Formula for the mean (binomial distribution):

N x P = Number of trials of an event x probability of the event occurring

We generally know that the Banker has a probability of 0.458597 while the Player has that of 0.446247, with Ties making up the difference.

For a shoe where the player wagered exclusively on Banker for 40 coups, the mean win for him would be:

40 x 0.458597 = 18.34388 (this means that he would be expected to win 18.34388 coups)

*the greater the difference between P and Q, the smaller the standard deviation!

Standard Deviation Of The Event Occurring

Here is a table calculating the number of coups a player wagering exclusively on Banker would win from a 40 coup shoe in terms of σ.

Total Coups

Mean

σ

-3 σ

-2 σ

-1 σ

1 σ

2 σ

3 σ

40

18.3438

3.1514

18.3438 – (3 x 3.1514) = 8.8896 coups

18.3438 – (2 x 3.1514) = 12.041 coups

18.3438 – (1 x 3.1514) = 15.1924 coups

18.3438 + (1 x 3.1514) = 21.4952 coups

18.3438 + (2 x 3.1514) = 24.6466 coups

18.3438 + (3 x 3.1514) = 27.798 coups

Now, to determine the expected result in dollars, we will multiply the player’s wager (assuming he wagered on ALL coups) by the coups he is expected to win or lose.

Determining Whether You Have a Losing Shoe

I have included 2 values, assuming average wagers of $1 and $5. You can add zeros to the backs of the average wagers as you please!

Percentage of Players Winning at that level

2.50%

13%

34%

50%

34%

13%

2.50%

Average Wager

Standard Deviations

-3 σ

-2 σ

-1 σ

Mean

1 σ

2 σ

3 σ

$1

Winning Coups

8.8896

12.041

15.1924

18.3438

21.4952

24.6466

27.798

Losing Coups

31.1104

27.959

24.8076

21.6562

18.5048

15.3534

12.202

Expected Result

($22.67)

($16.52)

($10.37)

($4.23)

$1.92

$8.06

$14.21

Percentage of Players Winning at that level

2.50%

13%

34%

50%

34%

13%

2.50%

Average Wager

Standard Deviations

-3 σ

-2 σ

-1 σ

Mean

1 σ

2 σ

3 σ

$5

Winning Coups

8.8896

12.041

15.1924

18.3438

21.4952

24.6466

27.798

Losing Coups

31.1104

27.959

24.8076

21.6562

18.5048

15.3534

12.202

Expected Result

($113.33)

($82.60)

($51.87)

($21.15)

$9.58

$40.30

$71.03

You can see that on average, players are expected to lose to the house. But when they start to win, the losing shoe is called – but at what point?

Notice the percentage levels above each σ. This means that for -2 σ, 13% of players would achieve a result of -$16.52, wagering at $1 a coup. At the mean, 50% of players would achieve a loss of -$4.23, wagering at $1 a coup.

To put that in perspective, a player wagering on Banker for 40 coups at $5,000 a coup would be expected to lose -$21,150. If that player starts to win more than $9,580, you might want to watch him more closely. A win of $40,300 would be unlikely and a win of $71,030, even more so. You would definitely want to call a losing shoe at that point!

The Z-Table

Now, the -3 to 3 σ measurement didn’t just pop out of nowhere. These measurements are derived from the Z-table.

Notice that the Z-table shows 0.0 to 3.0 down by the left and 0.00 to 0.09 on the top. This is the measurement in terms of σ, from 0.00 to 3.09.

You can determine the probability of σ by following the row on the left where the first 2 digits of your σ result appear to where it meets the column titled by the 3rd digit of your σ.

Examples:

0.16 σ would be 0.5636

1.82 σ would be 0.9656

The table is true for positive and negative σs, meaning that reading for -1.56 σ would be the same as 1.56 σ, which would be 0.9406.